Lets play a game.
We are two players and one dice.
I will toss first. If I get a 1, I win. If not, it is your turn.
You toss it, you win if you will get a 2. If not, it is my turn again.
We continue until someone wins.
What is the probability for me to win?
Recursively,
$$p =\frac{1}{6}+\left(\frac{5}{6}\right)\left(\frac{5}{6}\right)p$$
hence
$$p = \frac{6}{11}$$
Explanation: You win either right away, or else, if you get a chance to roll again, you're back to the initial state.
Alternatively, using Martin Rattigan's hint,
$$p =\frac{1}{6}+\left(\frac{5}{6}\right)(1-p)$$
which also yields
$$p = \frac{6}{11}$$
Explanation: You win either right away, or else, regarding the second player as the starting player in a new game, you want that player to lose.